Comprehensive Study Notes on Electrostatics, Gauss Law, and Atmospheric Electricity

Properties of Electric Charge

  • Fundamental Nature: Charge, like mass, is a fundamental and intrinsic property of matter.

  • Types of Charge: There are two types of charges: positive charge and negative charge.

    • Charge carried by a proton is positive.
    • Charge carried by an electron is negative.

  • Electrostatic Interactions:

    • Unlike charges attract each other.
    • Like charges repel each other.
    • The force driving these interactions is thought to result from the exchange of photons between the charged particles.

  • Conservation of Charge: Electric charge is always conserved.

  • Quantization of Charge: Charge is always in the form of an integral multiple of electronic charge and is never a fraction.

    • The mathematical representation is: q=±neq = \pm ne
    • Where nn is an integer and ee (the charge on an electron or proton) is the minimum charge, valued at: e=1.6×10−19 Ce = 1.6 \times 10^{-19}\,C

  • Additivity of Charge: Electric charge is additive. The total charge of a system is the algebraic sum of the individual charges.

  • Invariance of Charge: Electric charge is invariant, meaning it does not depend upon the motion of the charged body or the observer.

    • Mathematically: (q)at rest=(q)in motion(q)_{\text{at rest}} = (q)_{\text{in motion}}

Quarks

  • Origins/Discovery: Quarks were independently proposed by Gell-Mann and Zweig in 1963.

  • Etymology: They were named after a line in James Joyce's novel Three quarks for Muster Mark (referring to the three children of the character Mister/Muster Mark).

  • Definition: Quarks are truly elementary particles carrying fractional electronic charges.

  • Types (Flavours): There are six flavours of quarks:

    • Up (uu), Charm (cc), and Top or Truth (tt): All have a charge of +23e+\frac{2}{3}e.
    • Down (dd), Sideways or Strange (ss), and Bottom or Beauty (bb): All have a charge of −13e-\frac{1}{3}e.

  • Antiquarks: Designated by an overbar (e.g., uˉ\bar{u}, the anti-up quark), they possess charges opposite to their corresponding quarks. For example, uˉ\bar{u} has a charge of −23e-\frac{2}{3}e.

  • Composition of Hadrons:

    • Proton (uuduud): Composed of two up quarks and one down quark. Net charge: q=+23e+23e−13e=+eq = +\frac{2}{3}e + \frac{2}{3}e - \frac{1}{3}e = +e.
    • Neutron (uddudd): Composed of one up quark and two down quarks. Net charge: q=+23e−13e−13e=0q = +\frac{2}{3}e - \frac{1}{3}e - \frac{1}{3}e = 0.

  • Experimental Detection: Firm experimental evidence exists for all six quarks and their antiquarks within the nucleus, but free quarks have never been detected. Current theory implies it may be impossible to detect them directly in isolation.

Electric Field and Force Formulas

  • Electric Force: Electric force on charge q1q_1 due to charge q2q_2:     F=14πϵ0⋅q1q2(r−r1)3(r−r1)\mathbf{F} = \frac{1}{4\pi\epsilon_0} \cdot \frac{q_1q_2}{(\mathbf{r} - \mathbf{r}_1)^3} (\mathbf{r} - \mathbf{r}_1)

  • Field Strength (EE): Defined as the force per unit charge:     E=Fq\mathbf{E} = \frac{\mathbf{F}}{q}F=qE\mathbf{F} = q\mathbf{E}

    • Units: N/C\text{N/C}

  • Field of a Point Charge:     E=14πϵ0⋅qr2E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2}

  • General Vector Form (ii, jj, kk):     Ep=14πϵ0⋅q(distance)2\mathbf{E}_p = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{(\text{distance})^2}

  • Field Due to a Sphere of Charge (Radius RR):

    • Inside point (r≤Rr \le R): E=14πϵ0⋅qrR3E = \frac{1}{4\pi\epsilon_0} \cdot \frac{qr}{R^3}; thus E∝rE \propto r.
    • Outside point (r≥Rr \ge R): E=14πϵ0⋅qr2E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2}; thus E∝1r2E \propto \frac{1}{r^2}.
    • On the surface (r=Rr = R): E=14πϵ0⋅qR2E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{R^2}.

  • Field Due to a Hollow Sphere of Charge:

    • Inside (r<Rr < R): E=0E = 0
    • Outside (r≥Rr \ge R): E=14πϵ0⋅qr2E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r^2}
    • On the surface (r=Rr = R): E=14πϵ0⋅qR2=σϵ0E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{R^2} = \frac{\sigma}{\epsilon_0}

  • Field on the Axis of a Ring:

    • Value: E=14πϵ0⋅qx(R2+x2)32E = \frac{1}{4\pi\epsilon_0} \cdot \frac{qx}{(R^2 + x^2)^{\frac{3}{2}}}
    • At center (x=0x = 0): E=0E = 0
    • Far distance (x≫Rx \gg R): E=14πϵ0⋅qx2E = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{x^2}
    • As x→∞x \rightarrow \infty, E→0E \rightarrow 0

  • Infinitely Long Line Charge:     E=12πϵ0⋅λrE = \frac{1}{2\pi\epsilon_0} \cdot \frac{\lambda}{r}; E∝1rE \propto \frac{1}{r}

  • Thin Sheet of Charge:     E=σ2ϵ0E = \frac{\sigma}{2\epsilon_0}; this is constant and does not depend on distance (E∝r0E \propto r^0).

Electric Potential (VV)

  • Definition: The work done moving a charge from infinity to a point PP per unit charge:     VP=−W∞→PqV_P = -\frac{W_{\infty \rightarrow P}}{q}

  • Potential of a Point Charge:     V=14πϵ0⋅qrV = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r}

  • Potential Due to a Solid Sphere of Charge (Radius RR):

    • Inside (r≤Rr \le R): V=14πϵ0⋅qR3[1.5R2−0.5r2]V = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{R^3} [1.5R^2 - 0.5r^2]
    • Outside (r>Rr > R): V=14πϵ0⋅qrV = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r}

  • Potential Due to a Hollow Sphere:

    • Inside (r≤Rr \le R): Vinside=Vsurface=14πϵ0⋅qRV_{\text{inside}} = V_{\text{surface}} = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{R} (Constant)
    • Outside: V=14πϵ0⋅qrV = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{r}

  • Potential on the Axis of a Ring:     V=14πϵ0⋅qR2+r2V = \frac{1}{4\pi\epsilon_0} \cdot \frac{q}{\sqrt{R^2 + r^2}}

  • Work Done in Moving Charge:

    • By external agent: WA→B=q(VB−VA)W_{A \rightarrow B} = q(V_B - V_A)
    • By field: WA→B=q(VA−VB)W_{A \rightarrow B} = q(V_A - V_B)

  • Electrical Potential Energy (UU):

    • General: U=qVU = qV
    • Two point charges: U=kq1q2rU = \frac{k q_1 q_2}{r}
    • System of nn charges: Pair combinations equal to n(n−1)2\frac{n(n - 1)}{2}.

Electric Lines of Force

  • Definition: An imaginary line or curve through space where the tangent at any point indicates the direction of the electric vector (E\mathbf{E}). It represents the path a positive test charge would adopt independently.

  • Properties:

    1. Start from positive charges and terminate on negative charges.
    2. Exit positively charged conducting surfaces normal to the surface in an outward direction.
    3. Radial inwards for negative point charges; radial outwards for positive point charges.
    4. Always perpendicular to an equipotential surface.
    5. Exhibit longitudinal tension (contraction along length) and lateral pressure (expansion at right angles). Contraction denotes attraction; expansion denotes repulsion.
    6. The number of lines (flux) passing through unit normal area indicates electric intensity.
    7. Straight and radial for a charged sphere.
    8. Two lines of force never intersect.
    9. Parallel and equally spaced in a uniform field.

  • Relation between E\mathbf{E} and VV:

    • Vector form: E=−[∂V∂xi^+∂V∂yj^+∂V∂zk^]\mathbf{E} = -[\frac{\partial V}{\partial x}\mathbf{\hat{i}} + \frac{\partial V}{\partial y}\mathbf{\hat{j}} + \frac{\partial V}{\partial z}\mathbf{\hat{k}}]
    • Scalar form: E=−dVdrE = -\frac{dV}{dr}
    • Potential difference in uniform field: V=EdV = Ed

Electric Flux and Gauss Law

  • Electric Flux (\Phi_E):

    1. Perpendicular surface: ΦE=EA\Phi_E = EA
    2. Angled surface: ΦE=E⋅A=EAcos⁡(θ)\Phi_E = \mathbf{E} \cdot \mathbf{A} = EA\cos(\theta)
    3. Variable field: ΦE=∫E⋅dS\Phi_E = \int \mathbf{E} \cdot d\mathbf{S}

  • Gaussian Surface: A hypothetical closed surface used to calculate field intensity. Conditions for application:     a) Lines must be perpendicular to surface (θ=0∘\theta = 0^{\circ}).     b) Magnitude of the electric field must be equal at all points on the surface.

  • Gauss Theorem:     ∮E⋅ds=qnetϵ0\oint \mathbf{E} \cdot d\mathbf{s} = \frac{q_{\text{net}}}{\epsilon_0}

Electric Dipole

  • Definition: A combination of two equal and opposite charges separated by a small distance. It behaves as a single entity.

  • Dipole Moment: p=q(2l)\mathbf{p} = q(2\mathbf{l}). Direction is from negative to positive charge.

  • Field and Potential Calculations:

    • Axis: V≈kpr2V \approx \frac{kp}{r^2}; E=2kpr(r2−l2)2≈2kpr3E = \frac{2kpr}{(r^2 - l^2)^2} \approx \frac{2kp}{r^3}
    • Perpendicular Bisector: V=0V = 0; E=kp(r2+l2)32≈kpr3E = \frac{kp}{(r^2 + l^2)^{\frac{3}{2}}} \approx \frac{kp}{r^3}

  • Dipole in Uniform Electric Field:

    1. Net force (FnetF_{\text{net}}) = 0.
    2. Torque (τ\tau) = p×E=pEsin⁡(θ)\mathbf{p} \times \mathbf{E} = pE\sin(\theta).
    3. Potential Energy (UU) = −p⋅E=−pEcos⁡(θ)-\mathbf{p} \cdot \mathbf{E} = -pE\cos(\theta).
      • Stable Equilibrium (θ=0∘\theta = 0^{\circ}): U=−pEU = -pE (Minimum).
      • Unstable Equilibrium (θ=180∘\theta = 180^{\circ}): U=+pEU = +pE (Maximum).
    4. In a non-uniform field, a dipole undergoes both rotation and translation.

Van de Graaff Generator

  • Background: Designed by Van de Graaff in 1931 to produce high potential differences (millions of volts).

  • Operating Principles:

    1. Corona discharge (action of sharp points).
    2. Charge resides on the outer surface of a conductor. All charge given to a hollow conductor transfers to and distributes uniformly across the outer surface.

  • Applications: Accelerating charged particles (electrons, protons, ions) to high energies for artificial transmutation via target collision.

  • Potential Difference formula:     VA−VB=kq1r1−kq1r2=kq1(1r1−1r2)V_A - V_B = \frac{kq_1}{r_1} - \frac{kq_1}{r_2} = kq_1\left(\frac{1}{r_1} - \frac{1}{r_2}\right)     Where q1q_1 is charge on smaller sphere AA (r1r_1), and BB is the larger sphere (r2r_2).

Behavior of a Conductor in an Electrostatic Field

  1. Charge resides only on the outer surface.
  2. Electric field inside the conductor is zero.
  3. Electric potential inside is constant and equal to surface potential.
  4. Surface electric field is proportional to local surface density of charge (σ\sigma), but potential is independent of it.
  5. Outside field magnitude: E=σϵ0E = \frac{\sigma}{\epsilon_0}.
  6. Electric field inside a cavity is zero.
  7. Corona Discharge: Charge accumulates at sharp points (smallest radius of curvature). Leakage causes a faint glow (corona) and a hissing sound, often heard on power lines.

Charged Soap Bubble

  • Equilibrium conditions:

    • Inward Pressure (Surface Tension): PST=4TrP_{ST} = \frac{4T}{r}
    • Outward Pressure (Charging): PE=σ22ϵ0P_E = \frac{\sigma^2}{2\epsilon_0}
    • Equating pressures: 4Tr=σ22ϵ0\frac{4T}{r} = \frac{\sigma^2}{2\epsilon_0}

  • Charge formula:     q=8πr2rTϵ0q = 8\pi r \sqrt{2rT\epsilon_0}     (Assuming internal and external air pressures are equal).

Atmospheric Electricity

  • Definition: The study of electrical properties of the atmosphere during normal conditions and discharge (lightning).

  • Contributing Factors:

    1. Evaporation: Water vapor carries positive charge up, leaving water bodies negative.
    2. Ionization: Caused by UV radiation, cosmic rays, and radioactivity.
    3. Collisions: Ions produced cause further ionization via collision.
    4. Ionosphere: The region from 80 km80\text{ km} to 300 km300\text{ km} above Earth's surface.

  • Main Features:

    • Earth's field is 100 V/m100\text{ V/m} directed vertically downward. It decreases with height, becoming negligible at 50 km50\text{ km}.
    • Total potential difference (Surface to top): 400 kV400\text{ kV}.
    • Total Earth charge: −600 kC-600\text{ kC}.
    • Current density: 3.5×10−12 A/m23.5 \times 10^{-12}\,A/m^2.
    • Discharging current flowing to Earth: 1800 A1800\text{ A} (+1800 C/s+1800\text{ C/s} deposited).
    • The Earth would neutralize in 5 minutes5\text{ minutes} (300 s300\text{ s}), but constant thunderstorms/lightning keep the atmosphere charged.
    • Average lightning flash: Potential of 4×109 V4 \times 10^9\,V, charge of 15 C15\,C, energy of 2×1010 J2 \times 10^{10}\,J, and average upward current < 1 A1\,A.